Arc-regular cubic graphs of order four times an odd integer

  • Authors:

  • Marston D. Conder;Yan-Quan Feng

  • Affiliations:

  • Department of Mathematics, University of Auckland, Auckland, New Zealand 1142;Mathematics, Beijing Jiaotong University, Beijing, P.R. China 100044

  • Venue:
  • Journal of Algebraic Combinatorics: An International Journal
  • Year:
  • 2012

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Abstract

A graph is arc-regular if its automorphism group acts sharply-transitively on the set of its ordered edges. This paper answers an open question about the existence of arc-regular 3-valent graphs of order 4m where m is an odd integer. Using the Gorenstein---Walter theorem, it is shown that any such graph must be a normal cover of a base graph, where the base graph has an arc-regular group of automorphisms that is isomorphic to a subgroup of Aut(PSL(2,q)) containing PSL(2,q) for some odd prime-power q. Also a construction is given for infinitely many such graphs--namely a family of Cayley graphs for the groups PSL(2,p 3) where p is an odd prime; the smallest of these has order 9828.